The explicit solution to a sequential switching problem with non-smooth data

Abstract

We consider the problem faced by a decision maker who can switch between two random payoff flows. Each of these payoff flows is an additive functional of a general 1D Itô diffusion. There are no bounds on the number or on the frequency of the times at which the decision maker can switch, but each switching incurs a cost, which may depend on the underlying diffusion. The objective of the decision maker is to select a sequence of switching times that maximizes the associated expected discounted payoff flow. In this context, we develop and study a model in the presence of assumptions that involve minimal smoothness requirements from the running payoff and switching cost functions, but which guarantee that the optimal strategies have relatively simple forms. In particular, we derive a complete and explicit characterization of the decision maker's optimal tactics, which can take qualitatively different forms, depending on the problem data.

Keywords:

• optimal switching
• sequential entry and exit decisions
• stochastic impulse control
• system of variational inequalities

2000 Mathematics Subject Classifications::

• 93E20
• (49K45,60G40,91B28,91B70)

Acknowledgements

We are grateful to the organizers and the participants in the Optimal Stopping with Applications Symposium that was held at the University of Manchester on 22–27 January 2006, the Further Developments in Quantitative Finance workshop held at the ICMS in Edinburgh on 9–13 July 2007, and the Stochastic Filtering and Control Workshop that was held at the University of Warwick on 20–22 August 2007, for numerous helpful discussions. We also thank the anonymous referee whose comments helped improve the original version of the paper. Research supported by EPSRC grant nos. GR/S22998/01, EP/C508882/1.

Notes

^1. Email: [email protected]

Additional information

Notes on contributors

Timothy C. Johnson

1. 1. Email: [email protected]